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Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. (English) Zbl 1086.47057

The reviewer introduced and analyzed three-step iterative schemes for solving nonlinear operator equations in Hilbert spaces in [M. A. Noor, J. Math. Anal. Appl. 251, 217–229 (2000; Zbl 0964.49007)]. These three-step iterative schemese are also known as Noor iterations. These iterations have been extended and modified for several classes of nonexpansive mappings. It is well-known that Noor iterations include the Mann and Ishikawa iterations as special cases. B. Xu and M. A. Noor [J. Math. Anal. Appl. 267, 444–453 (2002; Zbl 1011.47039)] studied Noor iterations for a class of nonexpansive for asymptotically mappings.

In the paper under review, modified Noor-iterations are suggested and analyzed. In particular, the author investigates weak and strong convergence criteria of his modified Noor iterations under some suitable mild conditions. The results proved in this paper are interesting and present an improvement of the previously known results.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H05Monotone operators (with respect to duality) and generalizations